{
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   "source": [
    "using VMLS\n",
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chapter 15\n",
    "# Multi-objective least squares\n",
    "### 15.1 Multi-objective least squares\n",
    "Let’s write a function that solves the multi-objective least squares problem, with\n",
    "given positive weights. The data are a list (or array) of coefficient matrices (of\n",
    "possibly different heights) `As`, a matching list of (right-hand side) vectors `bs`, and\n",
    "the weights, given as an array or list, `lambdas`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "mols_solve (generic function with 1 method)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function mols_solve(As,bs,lambdas)\n",
    "k = length(lambdas);\n",
    "Atil = vcat([sqrt(lambdas[i])*As[i] for i=1:k]...)\n",
    "btil = vcat([sqrt(lambdas[i])*bs[i] for i=1:k]...)\n",
    "return Atil \\ btil\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Simple example.** We use the function `mols_solve` to work out a bi-criterion\n",
    "example similar to Figures [15.1](https://web.stanford.edu/~boyd/vmls/vmls.pdf#figure.15.1), [15.2](https://web.stanford.edu/~boyd/vmls/vmls.pdf#figure.15.2), and [15.3](https://web.stanford.edu/~boyd/vmls/vmls.pdf#figure.15.3) in VMLS. We minimize the weighted sum objective\n",
    "\n",
    "$$\n",
    "J_1 + λJ_2 = ‖A_1x− b_1‖^2 + λ‖A_2x− b_2‖^2\n",
    "$$\n",
    "\n",
    "for randomly chosen $10 × 5$ matrices $A_1, A_2$ and $10$-vectors $b_1$, $b_2$. The expression `lambdas = 10 .^ linspace(-4,4,200)` generates $200$ values of $λ ∈ [10^{−4}, 10^4]$, equally spaced on a logarithmic scale. The code creates the three plots in Figures [15.1](#fig_1), [15.2](#fig_2), and [15.3](#fig_3)."
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      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "As = [randn(10,5), randn(10,5)];\n",
    "bs = [randn(10), randn(10)];\n",
    "N = 200;\n",
    "lambdas = 10 .^ linspace(-4,4,200);\n",
    "x = zeros(5,N); J1 = zeros(N); J2 = zeros(N);\n",
    "for k = 1:N\n",
    "    x[:,k] = mols_solve(As, bs, [1, lambdas[k]])\n",
    "    J1[k] = norm(As[1]*x[:,k] - bs[1])^2\n",
    "    J2[k] = norm(As[2]*x[:,k] - bs[2])^2\n",
    "end;\n",
    "using Plots\n",
    "# plot solution versus lambda\n",
    "plot(lambdas, x', xscale = :log10, xlabel = \"lambda\")\n",
    "plot!(xlims = (1e-4,1e4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 15.1** Weighted-sum least squares solution $x̂(λ)$ as a function of $λ$ for\n",
    "a bi-criterion least squares problem with five variables.\n",
    "<a id=\"#fig_1\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "# plot two objectives versus lambda\n",
    "plot(lambdas, J1, xscale = :log10, label = \"J_1\")\n",
    "plot!(lambdas, J2, label = \"J_2\", xlabel = \"lambda\", xlims = (1e-4,1e4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 15.2** Objective functions $J_1 = ‖A_1x̂(λ)b_1‖^2$ (blue line) and $J_2 =\n",
    "‖A_2x̂(λ)b_2‖^2$ (red line) as functions of $λ$ for the bi-criterion problem in figure 15.1.\n",
    "<a id=\"#fig_2\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
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     "execution_count": 8,
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     "output_type": "execute_result"
    }
   ],
   "source": [
    "# plot trade-off curve\n",
    "plot(J1, J2, xlabel=\"J1\", ylabel = \"J2\", legend=false)\n",
    "# add (single-objective) end points to trade-off curve\n",
    "x1 = As[1] \\ bs[1];\n",
    "x2 = As[2] \\ bs[2];\n",
    "J1 = [norm(As[1]*x1-bs[1])^2, norm(As[1]*x2-bs[1])^2];\n",
    "J2 = [norm(As[2]*x1-bs[2])^2, norm(As[2]*x2-bs[2])^2];\n",
    "scatter!(J1,J2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 15.3** Optimal trade-off curve for the bi-criterion least squares problem\n",
    "of figures 15.1 and 15.2.\n",
    "<a id=\"#fig_3\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 15.2 Control\n",
    "### 15.3 Estimation and inversion\n",
    "**Estimating a periodic time series.** We consider the example of Figure [15.4](https://web.stanford.edu/~boyd/vmls/vmls.pdf#figure.15.4) in\n",
    "VMLS. We start by loading the data, as a vector with hourly ozone levels, for a\n",
    "period of $14$ days. Missing measurements have a value `NaN` (for Not a Number).\n",
    "The plot command skips those values (Figure [15.4](#fig_4)). "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
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   ],
   "source": [
    "ozone = ozone_data(); # a vector of length 14*24 = 336\n",
    "k = 14; N = k*24;\n",
    "plot(1:N, ozone, yscale = :log10, marker = :circle, legend=false)"
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   "metadata": {},
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    "**Figure 15.4** Hourly ozone level at Azusa, California, during the first $14$ days of July $2014$ (California Environmental Protection Agency, Air Resources Board, www.arb.ca.gov). Measurements start at $12$AM on July $1$st, and end at $11$PM on July $14$. Note the large number of missing measurements. In particular, all $4$AM measurements are missing.\n",
    "<a id=\"#fig_4\"></a>"
   ]
  },
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   "metadata": {},
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    "Next we use the `mols_solve` function to make a periodic fit, for the values\n",
    "$λ = 1$ and $λ = 10$. The Julia code `isnan` is used to find and discard the missing\n",
    "measurements. The results are shown in Figures [15.5](#fig_5) and [15.6](#fig_6)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
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     "execution_count": 14,
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     "output_type": "execute_result"
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   ],
   "source": [
    "A = vcat( [eye(24) for i = 1:k]...)\n",
    "# periodic difference matrix\n",
    "D = -eye(24) + [zeros(23,1) eye(23); 1 zeros(1,23)];\n",
    "ind = [k for k in 1:length(ozone) if !isnan(ozone[k])];\n",
    "As = [A[ind,:], D]\n",
    "bs = [log.(ozone[ind]), zeros(24)]\n",
    "# solution for lambda = 1\n",
    "x = mols_solve( As, bs, [1, 1])\n",
    "scatter(1:N, ozone, yscale = :log10, legend=false)\n",
    "plot!(1:N, vcat([exp.(x) for i = 1:k]...))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 15.5** Smooth periodic least squares fit to logarithmically transformed\n",
    "measurements, using $λ = 1$.\n",
    "<a id=\"#fig_5\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
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      ]
     },
     "execution_count": 15,
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     "output_type": "execute_result"
    }
   ],
   "source": [
    "# solution for lambda = 100\n",
    "x = mols_solve( As, bs, [1, 100])\n",
    "scatter(1:N, ozone, yscale = :log10, legend=false)\n",
    "plot!(1:N, vcat([exp.(x) for i = 1:k]...))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 15.6** Smooth periodic least squares fit to logarithmically transformed\n",
    "measurements, using $λ = 100$.\n",
    "<a id=\"#fig_6\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 15.4 Regularized data fitting\n",
    "**Example.** Next we consider the small regularized data fitting example of page [329](https://web.stanford.edu/~boyd/vmls/vmls.pdf#section*.367) of VMLS. We fit a model\n",
    "\n",
    "$$\n",
    "f̂(x) =\n",
    "\\sum^5_{k=1}\n",
    "θ_kf_k(x)\n",
    "$$\n",
    "\n",
    "with basis functions $f_1(x) = 1$ and $f_{k+1}(x) = sin(ω_kx + φ_k)$ for $k = 1, . . . , 4$ to $N = 20$ data points. We use the values of $ω_k$, $φ_k$ given in the text. We fit the model by solving a sequence of regularized least squares problems with objective\n",
    "\n",
    "$$\n",
    "\\sum^N_{i=1}(y^{(i)} − \\sum^5_{k=1} θ_kf_k(x^{(i)}))^2 + λ \\sum^5_{k=2}θ^2_k.\n",
    "$$\n",
    "\n",
    "The two plots are shown in Figures [15.7](#fig_7) and [15.8](#fig_8)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
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     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Import data as vectors xtrain, ytrain, xtest, ytest\n",
    "D = regularized_fit_data();\n",
    "xtrain = D[\"xtrain\"]; ytrain = D[\"ytrain\"];\n",
    "xtest = D[\"xtest\"]; ytest = D[\"ytest\"];\n",
    "N = length(ytrain);\n",
    "Ntest = length(ytest);\n",
    "p = 5;\n",
    "omega = [ 13.69; 3.55; 23.25; 6.03 ];\n",
    "phi = [ 0.21; 0.02; -1.87; 1.72 ];\n",
    "A = hcat(ones(N), sin.(xtrain*omega' + ones(N)*phi'));\n",
    "Atest = hcat(ones(Ntest),\n",
    "sin.(xtest*omega' + ones(Ntest)*phi'));\n",
    "npts = 100;\n",
    "lambdas = 10 .^ linspace(-6,6,npts);\n",
    "err_train = zeros(npts);\n",
    "err_test = zeros(npts);\n",
    "thetas = zeros(p,npts);\n",
    "for k = 1:npts\n",
    "    theta = mols_solve([ A, [zeros(p-1) eye(p-1)]],\n",
    "        [ ytrain, zeros(p-1) ], [1, lambdas[k]])\n",
    "    err_train[k] = rms(ytrain - A*theta);\n",
    "    err_test[k] = rms(ytest - Atest*theta);\n",
    "    thetas[:,k] = theta;\n",
    "end;\n",
    "using Plots\n",
    "# Plot RMS errors\n",
    "plot(lambdas, err_train, xscale = :log10, label = \"Train\")\n",
    "plot!(lambdas, err_test, xscale = :log10, label = \"Test\")\n",
    "plot!(xlabel = \"lambda\", ylabel = \"RMS error\", xlim = (1e-6, 1e6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 15.7** RMS training and test errors as a function of the regularization\n",
    "parameter $λ$.\n",
    "\n",
    "<a id=\"#fig_7\"></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Plot coefficients\n",
    "plot(lambdas, thetas', xscale = :log10)\n",
    "plot!(xlabel = \"lambda\", xlim = (1e-6, 1e6))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 15.8** The regularization path.\n",
    "<a id=\"#fig_8\"></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 15.5 Complexity\n",
    "**The kernel trick.** Let’s check the kernel trick, described in §[15.5.2](https://web.stanford.edu/~boyd/vmls/vmls.pdf#subsection.15.5.2), to find $x̂$, the minimizer of\n",
    "$$\n",
    "‖Ax− b‖^2 + λ‖x− x^{des}‖^2,\n",
    "$$\n",
    "where $A$ is an $m×n$ matrix and $λ > 0$. We’ll compute $x̂$ two ways. First, the naïve way, and then, using the kernel trick. We use the fact that if\n",
    "$$\n",
    "\\begin{bmatrix}\n",
    "A^T\\\\\n",
    "\\sqrt{λ}I\n",
    "\\end{bmatrix}\n",
    "= QR,\n",
    "$$\n",
    "then\n",
    "$$\n",
    "(AA^T + λI)^{−1} = (R^TQ^TQR)^{−1} = R^{−1}R^−T .\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 19.258670 seconds (40.15 k allocations: 1.130 GiB, 1.68% gc time)\n",
      "  0.558596 seconds (1.57 M allocations: 90.319 MiB, 4.78% gc time)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "1.1136057543636048e-13"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "m = 100; n = 5000;\n",
    "A = randn(m,n); b = randn(m); xdes = randn(n);\n",
    "lam = 2.0;\n",
    "# Find x that minimizes ||Ax-b||^2 + lambda ||x||^2\n",
    "@time xhat1 = [A; sqrt(lam)*eye(n)] \\ [b; sqrt(lam)*xdes];\n",
    "# Now use kernel trick\n",
    "@time begin\n",
    "Q, R = qr([A' ; sqrt(lam)*eye(m)]);\n",
    "Q = Matrix(Q);\n",
    "xhat2 = A' * (R \\ (R' \\ (b-A*xdes))) + xdes;\n",
    "end;\n",
    "norm(xhat1-xhat2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The naïve method requires the factorization of a $5100 × 5100$ matrix. In the second\n",
    "method we factor a matrix of size $5100 × 100$."
   ]
  }
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